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REVIEW 3 major objections 4 minor 35 references

A proof of Perrin-Riou's Heegner point main conjecture

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Pith's one-line read The Heegner point main conjecture is proved under mild hypotheses.

desk verdict A serious proof of Perrin-Riou's Heegner point main conjecture that removes old restrictions; the stated theorem is conditional on non-anomalous p, a limitation the authors themselves flag as removable. read the letter →

arxiv 1908.09512 v2 pith:OL7AYYRK submitted 2019-08-26 math.NT

Ashay Burungale , Francesc Castella , Chan-Ho Kim This is my paper · ORCID
classification math.NT MSC 11R2311F33
keywords IwasawatheoryHeegnerpointsbipartiteEulersystemsanticyclotomicZ_p-extensionsp-adicL-functionsellipticcurveslevelraising
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves the Heegner point main conjecture, an Iwasawa-theoretic statement formulated in 1987 that predicts the size of an elliptic curve's Tate--Shafarevich group over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field is governed by a system of Heegner points. The main theorem establishes this equality of characteristic ideals for every elliptic curve over $\mathbb{Q}$ with good ordinary reduction at a prime $p>3$, for every imaginary quadratic field satisfying the Heegner hypothesis, provided a mild ramification condition, surjectivity of the mod-$p$ Galois representation, and a non-anomaly condition on $p$ hold. It removes two restrictions of earlier work: the conductor need not be squarefree and $p$ may be inert in $K$. A second theorem, in the split case, derives the Iwasawa--Greenberg main conjecture for the anticyclotomic $p$-adic $L$-function associated to the curve, so the two conjectures are proved to be equivalent forms of one statement.

What carries the argument

The load-bearing object is a bipartite Euler system: a pair of compatible families $\kappa_j(m)$ and $\lambda_j(m)$ indexed by squarefree products of admissible primes (primes inert in $K$ and satisfying a congruence condition), obtained by level-raising the $p$-stabilized newform at those primes. The $\lambda$-side consists of elements in the Iwasawa algebra modulo $\wp^j$, and a criterion from the bipartite Euler system method says the main conjecture's divisibility becomes equality if the $\lambda$-system has nonzero image modulo $\wp$. The paper proves this nonzero image by induction on the Selmer rank: level-raising at two admissible primes lowers the rank by two, and in the rank-one base case the non-anomalous hypothesis makes a certain $p$-adic multiplier nonzero, so the interpolated algebraic $L$-value detects the system. In short, the proof upgrades primitivity of the Heegner-point system to the $\Lambda$-level primitivity that forces equality of characteristic ideals.

What would settle it

For a chosen triple satisfying every hypothesis, compute at the trivial character the interpolation formula $\lambda_1(q)^2 \equiv e_p(g,1)\,L_{\mathrm{alg}}(g/K)$ modulo $\wp$ for a level-raising prime $q$; the theorem predicts the right-hand side is a $p$-adic unit for some $q$. Finding a triple where the right-hand side vanishes modulo $\wp$ for every admissible $q$, while the $\wp$-Selmer rank is at least three, would break the proposition that gives the nonzero image, and with it the proof of the main theorem.

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Extended reading notes

Core claim

Theorem A is the central claim: under the stated hypotheses, the Pontryagin dual of the Selmer group and the projective limit of $p$-adic Selmer groups both have $\Lambda$-rank one, and the torsion module $M_\infty$ satisfies $\operatorname{Char}_\Lambda(M_\infty)=\operatorname{Char}_\Lambda(S_\infty/\Lambda\kappa_\infty)$ together with the self-duality $\operatorname{Char}_\Lambda(M_\infty)=\operatorname{Char}_\Lambda(M_\infty)^\iota$. In concrete terms, the $p$-primary Tate--Shafarevich group over the tower is measured exactly, up to pseudo-isomorphism, by the Iwasawa module generated by Heegner points. The theorem also yields a $p$-converse: if the Selmer group has corank one, the analytic rank of $L(E/K,s)$ is one. When $p$ splits in $K$, the same proof establishes that the square of the relevant $p$-adic $L$-function generates the characteristic ideal of a modified Selmer group, i.e. the Iwasawa--Greenberg main conjecture.

Load-bearing premise

The argument depends on $p$ being non-anomalous for $E$ over $K$: $p$ divides neither $|\tilde E(\mathbb{F}_w)|$ for either prime $w$ of $K$ above $p$. This is what makes the $p$-adic multiplier $e_p(g,1)$ nonzero, and without it the proof that the $\lambda$-system has nonzero image modulo $\wp$ does not go through.

Editorial extensions

If this is right

  • The Heegner point main conjecture now holds for all triples satisfying the stated hypotheses, including the previously excluded cases $N^-=1$ and $p$ inert in $K$.
  • The $p$-converse to the analytic-rank-one theorem follows without invoking the structure theorem for Selmer groups: corank-one Selmer implies $L(E/K,s)$ has a simple zero at $s=1$.
  • In the split case, the Iwasawa--Greenberg main conjecture holds for the anticyclotomic $p$-adic $L$-function: its square generates the characteristic ideal of the dual of the modified Selmer group.
  • The two main conjectures are equivalent under the stated hypotheses, so any future control theorem or divisibility for one transfers to the other.
  • The divisibility obtained earlier by the bipartite Euler system method is upgraded to an equality, so the upper and lower bounds on Selmer growth coincide.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: the non-anomalous hypothesis appears removable, since the rank-one appendix already proves the same conclusions without it; the same induction should yield the main conjecture under the ramification and surjectivity hypotheses alone if the multiplier obstruction can be bypassed.
  • Extension: the primitivity mechanism is not tied to this particular curve; any setting in which a bipartite Euler system is built from level-raising and a nonvanishing $p$-adic $L$-value could receive the same 'primitivity implies equality' upgrade.
  • Extension: a concrete numerical check in a small-conductor example could test the mechanism independently of full characteristic-ideal computations, since the theory predicts that the rank-one $\lambda$-system has a $p$-adic unit image.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proves Perrin-Riou's Heegner point main conjecture for an elliptic curve E/Q with good ordinary reduction at p > 3 and an imaginary quadratic field K satisfying the generalized Heegner hypothesis, under the assumptions that Hypothesis ♠ holds, the residual representation ρ is surjective, and p is non-anomalous. The proof combines Howard's theory of bipartite Euler systems with Wei Zhang's proof of Kolyvagin's conjecture; the key step (Proposition 3.7) is to show that the associated λ-system has nonzero image modulo ℘, which triggers Howard's equality criterion. When p splits in K, the authors also derive the Iwasawa–Greenberg main conjecture for the Bertolini–Darmon–Prasanna p-adic L-function, via an explicit reciprocity law (Theorem 4.4) and an equivalence between the two main conjectures (Theorem 5.2). An appendix gives a rank-one proof that avoids the non-anomalous hypothesis in the split, analytic-rank-one case.

Significance. If correct, this is a major advance in anticyclotomic Iwasawa theory: it establishes the Heegner point main conjecture for a broad class of elliptic curves, including cases where N^- = 1, where N is not squarefree, and where p is inert in K, all of which were previously inaccessible. The paper also supplies the missing details in the literature for the explicit reciprocity law for N^- ≠ 1, and shows the equivalence between the Heegner point and Iwasawa–Greenberg main conjectures in the split case. The argument is a substantial synthesis of deep external results (Howard, W. Zhang, Cornut–Vatsal, Chida–Hsieh, Manning–Shotton), and the paper is careful to state its hypotheses and to identify the role of each input. The proof is not circular: it derives the conjecture from Howard's machinery and W. Zhang's theorem, and the use of the authors' earlier work [CH18a] is a legitimate prior result, not an assumption of the conjecture.

major comments (3)
  1. [§3, Proposition 3.7] The proof of Proposition 3.7 is not written in full at the load-bearing step. The case r = 1 is explained, but for r ≥ 3 the argument is summarized as 'by the argument in the proof of Theorem 9.1 we can find a form g2 ... with associated Selmer rank equal to r − 2', and then the induction hypothesis is applied without verifying that the p-adic multiplier e_p(g,1) remains nonzero at every descent stage. Since the nonvanishing of λ modulo ℘ is exactly the input that triggers Howard's equality criterion in Theorem 3.4, this is a central point. The paper's own Remark A.7 and Appendix A show that the non-anomalous assumption can be removed only in the split analytic-rank-one case, so Theorem A remains conditional on an arithmetic condition whose scope in higher rank is not resolved. This is not an internal inconsistency, but the main theorem's proof needs a complete induction argument or an explicit statement that the higher-rank case is deferred.
  2. [§4.2, Theorem 4.4] The explicit reciprocity law for N^- ≠ 1 is a key ingredient in the proof of Theorem B, and the introduction notes that the details for this extension were previously missing in the literature. However, the proof given here says only that after replacing Proposition 3.3 by Proposition 4.1, 'the argument in [CH18a, Thm. 5.7] applies verbatim'. The paper should provide a full verification of the Shimura-curve-to-Igusa comparison and of the formal-group logarithm computation in the general N^- case, rather than leaving this to a verbatim-reference claim. Without these details, the equivalence in Theorem 5.2 is not self-contained.
  3. [§3, Theorem 3.3 and Lemma 3.6] The construction of the bipartite Euler system (κ, λ) is summarized via references to Chida–Hsieh and Bertolini–Darmon, but the verification that the hypotheses of Howard's theory, such as [How06, Hyp. 2.2.4 and 2.3.1], hold in this setting is not carried out. Lemma 3.6 asserts that Hypothesis 2.3.1 holds by [BD05, Thm. 3.2], and the existence of core vertices is invoked from [How06, Cor. 2.4.9], but the reader cannot check from the present text that all technical conditions (e.g., the admissibility index and the ordinary local conditions at primes dividing N^- m) are satisfied uniformly for the systems constructed in Theorem 3.3. This is a necessary part of the application of Howard's criterion and should be spelled out.
minor comments (4)
  1. [§2, p. 5] The phrase 'f is assumed to be ordinarity at p' should read 'f is assumed to be ordinary at p'.
  2. [§3, Proposition 3.7] The sentence 'the image of λ_1(q)^2 under the augmentation map ... is given by e_p(g,1)·L_alg(g/K) (mod ℘) up to a p-adic unit' should specify the normalization of the isomorphism H^1_ord(K_q, T_j) ≅ Λ/℘^jΛ, since a different normalization changes the unit but not the nonvanishing claim; this would help readers verify the congruence.
  3. [§4.2, Corollary 4.5] The nonvanishing of loc_p(κ∞) is derived from the implication κ_χ ≠ 0 ⇒ loc_v(κ_χ) ≠ 0, but the cited [Nek07, Thm. 3.2] is used in a way that should be briefly justified: namely, that the Heegner-class construction gives a point whose residue class is nonzero when the global class is nonzero. A short explanation would improve readability.
  4. [§1, Theorem A] The phrase 'under mild hypotheses' in the abstract and introduction undersells the non-anomalous condition, which is a substantive arithmetic hypothesis on (E,p,K) and is not removed in the main theorem except in the special case of Appendix A; consider rephrasing to 'under explicit hypotheses'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation rests on independent external results and explicit hypotheses, not on the conjecture being proved.

full rationale

The paper's central claim (Theorem A / Theorem 3.2) is not assumed as an input. The proof combines Howard's bipartite Euler system criterion (Theorem 3.4) with a nonvanishing statement for the lambda system (Proposition 3.7). Theorem 3.4 is an external result of Howard, and Proposition 3.7 is proved using W. Zhang's proof of Kolyvagin's conjecture [Zha14] together with level-raising results of Chida--Hsieh [CH15]. The bipartite Euler system in Theorem 3.3 is constructed from Heegner points and level-raised forms, not from the characteristic ideal equality that is being derived. The hypotheses used—Hypothesis ♠, surjectivity of ρ, and non-anomalousness of p—are explicit arithmetic conditions that guarantee the relevant level-raising and that the p-adic multiplier e_p(g,1) is nonzero; they do not encode the conclusion. The non-anomalous assumption is stated as a hypothesis and is discussed in Remark A.7 as removable in the rank-one case, which is a limitation of the proof rather than a circularity. The paper does cite prior work by one of the present authors, notably [CH18a] (Castella--Hsieh) for the explicit reciprocity law and [Cas17] for the Selmer-group comparison underlying Theorem 5.2. These are published, parameter-free results whose stated assumptions do not include Perrin-Riou's conjecture; moreover, the paper supplies the missing N^- ≠ 1 extension of the reciprocity law instead of merely assuming it. Under the reviewing rules, such independent support does not raise the circularity score. No equation in the paper reduces by definition to the target equality, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data, and no new entities are introduced. The proof is a deductive chain whose load-bearing external inputs are listed. The most important are Zhang's proof of Kolyvagin's conjecture (used to prove primitivity of the Euler system) and Howard's bipartite Euler system criterion. The auxiliary choices in the construction (an auxiliary prime ℓ0 and an auxiliary Hecke character ψ) are standard and do not affect the result.

assumptions (6)
  • domain assumption Wei Zhang's proof of Kolyvagin's conjecture (Zha14), specifically the primitivity of Heegner point Kolyvagin systems.
    Proposition 3.7 proves the nonvanishing of λ mod ℘ by induction on the Selmer rank, citing Theorem 7.2 and the argument of Theorem 9.1 in Zha14. The central equality in Theorem A depends on this.
  • domain assumption Howard's theory of bipartite Euler systems (How06), including the criterion for equality in Theorem 3.4.
    The entire proof of Theorem A rests on Howard's framework and the criterion that divisibility becomes equality when the lambda system is primitive. This is cited as Theorem 3.4.
  • domain assumption Cornut-Vatsal nonvanishing of Heegner point classes (CV07).
    Used in Theorem 3.4 and Corollary 4.5 to show κ∞ is nonzero and S has Λ-rank one.
  • domain assumption Ihara's lemma for Shimura curves as proved by Manning-Shotton (MS20).
    Used in the construction of the bipartite Euler system in Theorem 3.3 to remove the earlier non-anomalous requirement in the level-raising step.
  • domain assumption Castella-Hsieh explicit reciprocity law (CH18a), extended in this paper to N^-≠1.
    Theorem 4.4 and Theorem 5.2 rely on this reciprocity law to relate κ∞ to L^BDP_p and to prove the equivalence of the two main conjectures.
  • domain assumption Gross-Zagier formula (GZ86, YZZ13).
    Used in Appendix A to relate nonvanishing of L'(E/K,1) to the Heegner point having infinite order in the alternative rank-one approach.

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Pith. "Pith review of A proof of Perrin-Riou's Heegner point main conjecture." pith.science (2026). https://pith.science/paper/OL7AYYRK

@misc{pith2026190809512,
  author       = {Pith},
  title        = {Pith review of: A proof of Perrin-Riou's Heegner point main conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OL7AYYRK}},
  note         = {Machine review of arXiv:1908.09512}
}
abstract

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate-Shafarevich group of $E$ over the anticyclotomic $\mathbf{Z}_p$-extension of $K$ in terms of Heegner points. In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when $p$ splits in $K$, we also obtain a proof of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna.

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